This is correct as far as I know. Inner city public school funding is comparable to some private school funding in some locations and higher than funding of some parochial school funding.. Perhaps inner city public school funding is even higher then the mean of all private school funding. We should consider, however, that that mean for private schools is watered down by low funding of a lot of very mediocre private schools in the Deep South that sprang up after integration.
If you were a teacher and offered a job at Choate versus a Detroit inner city school, how much premium in salary would you require to except the Detroit job over the Choate job? So it clearly isn't just a simple metric like funding that makes the difference; it is what you do with the funding, and a host of other factors. It is a multivariate problem that can't be addressed effectively by just throwing money at it. A kid who does poorly either academically or socially, as a rule, won't be in a good private school very long, and odds are poor that they would even be there in the first place, if we are considering the better and most selective private schools.
There is a lack of sufficient alternatives in many public school systems, and a total lack, naturally, in the private schools. The need for educational alternatives is unique to the public school systems, and dealing with students that fall outside the mainstream is a cost that private schools don't have to consider. We can expect a good public school system to provide not only for mainstream students, but also for the educationally challenged --what my generation called retarded students; for those students with a greater interest in vocational education, and for students with behavioral problems requiring close supervision and special programs. Providing these alternatives to public school students is expensive if done well. This is something the private schools don't deal with.
Oversimplifying a complex problem will lead easily to ineffective solutions. We can deal well with all of these problems inherent in public education, but not if we try to do it in the political arena.
I appreciate your detailed comment but I think you deflect from some key simple realities. First, is the fact that inner-city schools are generally well funded by any measure.
Inner-City Schools spend more per student than state average spending and generally more than the top performing suburban schools. Inner-City spending is comparable to the best private school tuitions, much higher than average privates, and much, much higher than religious privates.
For examples, Camden, NJ (one of worst school systems in the U.S.) spent about $27,000 per student in 2015, NJ public average was about $18,500 (generally among best public schools in U.S.); Milburn/Short Hills, NJ spent $19,000...one of the top school districts in the U.S. Tuition at Delbarton School or Pingry, private schools ranked in top of all U.S. Schools, is about $25,000 to $32,000 depending on grade.
So, generally speaking, the worst school in the U.S. costs in the ball park as the best school in the U.S.
Second, Public Schools are political by nature; as you suggest that is a large part of why the issues are so intractable.
Third, the teacher salaries are the key expense issue; your Choate/Detroit analogy is deceptive because it assumes that it is.
The key expense issues are administrative budgets and administrative salaries, the ratio of administrators to teachers, along with cost of maintenance of facilities, maintenance staffing and salaries, and significantly, with employees, the excessive cost of medical insurance and defined benefit pensions that are gamed.
The real tragedy in public school funding today is that all increased revenue from hear on out will likely be applied to pension and medical costs for teachers who are retired and have moved away from the school districts that must continue to pay them. Actual funding for education is decreasing as more revenue is allocated to the unfunded entitlement problem.
Finally, with regard to the argument that public schools must be maintained because of special needs and low performing students; that argument fails on multiple levels.
First, this is no excuse to under serve the needs of the majority of students, just to serve the special needs of a minority. You can do both, and they don't need to be done together in all cases and they don't need to funded by the same budget or from the same revenue sources. You conflate a social obligation and duty with a general education objective. The issue narrative was positioned that way politically and it has hurt the education system as a whole.
Second, it is not true that private schools positively select leaving the public school with the job of educating the worst students. The prevalent private school presence in the inner-city is the parochial school system which accepts all students and is generally not merit based. Studies show that students of the same underprivileged back ground do better in the private environment than the public school system, not because they are positively selected. There is something wrong with the public schools that is not caused by the students.